Journal of Petroleum Science and Engineering 78 (2011) 582–585

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Journal of Petroleum Science and Engineering

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The flow of power-law fluids in axisymmetric corrugated tubes

Taha SochiImaging Sciences & Biomedical Engineering, King's College London, The Rayne Institute, St Thomas' Hospital, London, SE1 7EH, UK

E-mail address: taha.sochi@kcl.ac.uk.

0920-4105/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.petrol.2011.08.006

a b s t r a c t

a r t i c l e i n f o

Article history:Received 9 November 2010Accepted 8 August 2011Available online 19 August 2011

Keywords:Non-newtonian flowPower-law fluidCorrugated tubeAxisymmetric

In this article we present an analytical method for deriving the relationship between the pressure drop andflow rate in laminar flow regimes, and apply it to the flow of power-law fluids through axially-symmetriccorrugated tubes. Themethod, which is general with regard to fluid and tube shapewithin certain restrictions,can also be used as a foundation for numerical integration where analytical expressions are hard to obtain dueto mathematical or practical complexities. Five converging–diverging geometries are used as examples toillustrate the application of this method.

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© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Modeling the flow in corrugated tubes is required for a number oftechnological and industrial applications. Moreover, it is a necessarycondition for correct description of several phenomena such as yield-stress, viscoelasticity and the flow of Newtonian and non-Newtonianfluids through porous media (Pilitsis and Beris, 1989; Sochi, 2009,2010a, 2010b, 2010c, 2010d; Talwar and Khomami, 1992). In theliterature of fluid mechanics there are numerous studies on the flowthrough tubes or channels of corrugated nature. Most of these studiesuse numerical techniques such as spectral and finite differencemethods (Burdette et al., 1989; James et al., 1990; Lahbabi andChang, 1986; Momeni-Masuleh and Phillips, 2004; Phan-Thien andKhan, 1987; Phan-Thien et al., 1985; Pilitsis and Beris, 1989, 1992).Some others adopt analytical approaches based on simplifiedassumptions and normally deal with very special cases (Oka, 1973;Williams and Javadpour, 1980). The current paper presents amathematical method for deriving analytical relations between thevolumetric flow rate and pressure drop in corrugated circularcapillaries of variable cross section, such as those illustratedschematically in Fig. 1. It also presents five examples in which thismethod is used to derive flow equations for non-Newtonian fluids ofpower-law rheology. The method, however, is more general and canbe used for other tube shapes and other rheologies. The followingderivations assume a laminar flow of a purely-viscous incompressiblefluid where the tube corrugation is smooth and relatively small toavoid complex flow phenomena which are not accounted for in theunderlying assumptions of this method.

2. P–Q relation for power-law fluids

The power-law, or Ostwald-deWaele, is a widely-used fluid modelto describe the rheology of shear thinning fluids. It is one of thesimplest time-independent non-Newtonian models as it contains twoparameters only. The model is given by the relation (Bird et al., 1987;Carreau et al., 1997)

μ =τγ̇

= Cγ̇ n−1 ð1Þ

where μ is the fluid viscosity, τ is the stress, γ̇ is the strain rate, C is theconsistency factor, and n is the flow behavior index. In Fig. 2 the bulkrheology of this model for shear-thinning case is presented genericallyon log–log scales. Although the power-law is used to model shear-thinning fluids, it can also be used for modeling shear-thickening bymakingn greater thanunity. Themajorweakness of power-lawmodel isthe absence of plateaux at low and high strain rates. Consequently, itfails to produce sensible results in these flow regimes.

The volumetric flow rate, Q, of a power law fluid through a circularcapillary of constant radius r with length x across which a pressuredrop P is applied is given by

Q =πnr4P1=n

2xC1=n 3n + 1ð Þ2xr

� �1−1=n: ð2Þ

A derivation of this relation, which can be obtained from Eq. (1),can be found in Appendix A. On solving this equation for P thefollowing relation is obtained

P =2CQn 3n + 1ð Þnx

πnnnr3n + 1 : ð3Þ

Fig. 1. Profiles of converging–diverging axisymmetric capillaries.

→Rmax Rmin Rmax

↑

x0−L/2 L/2

r

Fig. 3. Schematic representation of the radius of a conically shaped converging–diverging capillary as a function of the distance along the tube axis.

583T. Sochi / Journal of Petroleum Science and Engineering 78 (2011) 582–585

For an infinitesimal length, δx, of a capillary, the infinitesimalpressure drop for a given flow rate Q is

δP =2CQn 3n + 1ð Þnδx

πnnnr3n + 1 : ð4Þ

For an incompressible fluid, the volumetric flow rate across anarbitrary cross section of the capillary is constant. Therefore, the totalpressure drop across a capillary of length L with circular cross sectionof varying radius, r(x), is given by

P =2CQn 3n + 1ð Þn

πnnn ∫ L

0

dxr3n + 1 : ð5Þ

In the following sections, Eq. (5) will be used to derive analyticalexpressions for the relation between pressure drop and volumetricflow rate for five converging–diverging geometries of axially-symmetric pipes. The method can be equally applied to othergeometries as long as the tube radius can be expressed analyticallyas a function of the tube axial distance.

2.1. Conic tube

For a corrugated tube of conic shape, depicted in Fig. 3, the radius r asa function of the axial coordinate x in the designated frame is given by

r xð Þ = a + bjxj −L = 2 ≤ x ≤ L = 2 a; b > 0 ð6Þ

Strain Rate (s−1)

Vis

cosi

ty (

Pa.s

)

Slope = − 1

Fig. 2. The bulk rheology of power-law fluids on logarithmic scales for finite strain rates(γ̇ N0).

where

a = Rmin and b =2 Rmax−Rminð Þ

L: ð7Þ

Hence, Eq. (5) becomes

P =2CQn 3n + 1ð Þn

πnnn ∫ L = 2

−L = 2

dxa + bjxjð Þ3n + 1 ð8Þ

=2CQn 3n + 1ð Þn

πnnn1

3bn a−bxð Þ3n" #

0

−L =2

+ − 13bn a + bxð Þ3n

" #L=20Þ0

@ ð9Þ

=4CQn 3n + 1ð Þn

3πnnn + 1b1a3n

− 1a + bL=2ð Þ3n

" #ð10Þ

that is

P =2LCQn 3n + 1ð Þn

3πnnn + 1 Rmax−Rminð Þ1

R3nmin

− 1R3nmax

" #: ð11Þ

2.2. Parabolic tube

For a tube of parabolic profile, depicted in Fig. 4, the radius is givenby

r xð Þ = a + bx2 −L= 2 ≤ x ≤ L = 2 a; b > 0 ð12Þ

where

a = Rmin and b =2L

� �2Rmax−Rminð Þ: ð13Þ

Therefore, Eq. (5) becomes

P =2CQn 3n + 1ð Þn

πnnn ∫ L = 2

−L = 2

dx

a + bx2� �3n + 1 ð14Þ

=2CQn 3n + 1ð Þn

πnnnax

bx2

a + 1a + bx2

!3n

2F112;3n + 1;

32;− bx2

a

!24

35L = 2

−L = 2

ð15Þ

→Rmax Rmin Rmax

↑

x0−L/2 L/2

r

Fig. 4. Schematic representation of the radius of a converging–diverging capillary with aparabolic profile as a function of the distance along the tube axis.

→Rmax Rmin Rmax

↑

x0−L/2 L/2

r

Fig. 5. Schematic representation of the radius of a converging–diverging capillary with asinusoidal profile as a function of the distance along the tube axis.

584 T. Sochi / Journal of Petroleum Science and Engineering 78 (2011) 582–585

where 2F1 is the hypergeometric function. Therefore

P =2LCQn 3n + 1ð Þn

nnπnR3n + 1min

2F112;3n + 1;

32;1−Rmax

Rmin

� �: ð16Þ

2.3. Hyperbolic tube

For a tube of hyperbolic profile, similar to the profile in Fig. 4, theradius is given by

r xð Þ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia + bx2

p−L = 2 ≤ x ≤ L= 2 a; b > 0 ð17Þ

where

a = R2min and b =

2L

� �2R2max−R2

min

� �: ð18Þ

Therefore, Eq. (5) becomes

P =2CQn 3n + 1ð Þn

πnnn ∫ L = 2

−L = 2

dx

a + bx2� � 3n + 1ð Þ=2 ð19Þ

=2CQn 3n + 1ð Þn

πnnn xbx2

a + 1a + bx2

!3n + 12

2F112;3n + 1

2;32− bx2

a

!" #L = 2

−L = 2

ð20Þ

where 2F1 is the hypergeometric function. Therefore

P =2LCQn 3n + 1ð Þn

πnnnR3n + 1min

2F112;3n + 1

2;32; 1−R2

max

R2min

!: ð21Þ

2.4. Hyperbolic cosine tube

For a tube of hyperbolic cosine profile, similar to the profile inFig. 4, the radius is given by

r xð Þ = a cosh bxð Þ −L = 2 ≤ x ≤ L = 2 a > 0 ð22Þ

where

a = Rmin and b =2Larccosh

Rmax

Rmin

� �: ð23Þ

Hence, Eq. (5) becomes

P =2CQn 3n + 1ð Þnπnnna3n + 1 ∫ L = 2

−L = 2

dxcosh3n + 1 bxð Þ ð24Þ

=2CQn 3n + 1ð Þn3πnnn + 1a3n + 1b

sinh bxð Þcosh3n bxð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−sinh2 bxð Þ

q 2F112;−3n

2;2−3n

2; cosh2 bxð Þ

� �264

375L = 2

−L = 2

:

ð25Þ

On evaluating the last expression and taking its real part which isphysically significant, the following relation is obtained

P =2LCQn 3n + 1ð Þn

3πnnn + 1RminR3nmax arccosh

Rmax

Rmin

� � Im 2F112;−3n

2;2−3n

2;R2max

R2min

! !

ð26Þ

where Im(2F1) is the imaginary part of the hypergeometric function.

2.5. Sinusoidal tube

For a tube of sinusoidal profile, depicted in Fig. 5, where the tubelength L spans one complete wavelength, the radius is given by

r xð Þ = a−b cos kxð Þ −L= 2 ≤ x ≤ L = 2 a > b > 0 ð27Þ

where

a =Rmax + Rmin

2b =

Rmax−Rmin

2& k =

2πL

ð28Þ

Hence, Eq. 5 becomes

P =2CQn 3n + 1ð Þn

πnnn ∫ L = 2

−L = 2

dxa−b cos kxð Þ½ �3n + 1 ð29Þ

= −2CQn 3n + 1ð Þn3πnnn + 1k

1sin kxð Þ a−b cos kxð Þ½ �3n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−sin2 kxð Þa2−b2

sA

24

35L = 2

−L = 2

ð30Þ

where A is the Appell hypergeometric function given by

A = F1 −3n;12;12;1−3n;

a−b cos kxð Þa + b

;a−b cos kxð Þ

a−b

� �: ð31Þ

On taking the limit as x→− L2

þand x→

L2

−and considering the real

part, the following relation is obtained

P =2LCQn 3n + 1ð Þn

3πn + 1nn + 1R3nmax

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRmaxRmin

p Im F1 −3n;12;12;1−3n;1;

Rmax

Rmin

� �� �

ð32Þ

where Im(F1) is the imaginary part of the Appell hypergeometricfunction.

It is noteworthy that all these relations (i.e. Eqs. (11), (16), (21),(26) and (32)), are dimensionally consistent. Moreover, they havebeen thoroughly tested and validated by numerical integration.

3. Conclusions

In this article an approximate mathematical method for obtaininganalytical relations between the flow rate and pressure drop inaxisymmetric corrugated tubes is presented and applied to the flow ofpower-law fluids. The method is illustrated by five examples ofcircular capillaries with converging–diverging shape. This method canbe used to derive similar relations for other types of fluid and othertypes of tube geometry. It can also be used as a base for numericalintegration where analytical relations are difficult to obtain due tomathematical or practical reasons. There are many scientific andindustrial applications that can benefit from the use of the derivedexpressions and the underlying method as a good approximation invarious practical situations such as modeling the flow in non-uniformducts or porous media.

585T. Sochi / Journal of Petroleum Science and Engineering 78 (2011) 582–585

Nomenclatureγ̇ strain rate (s−1)μ fluid viscosity (Pa.s)τ stress (Pa)C consistency factor (Pa.sn)F1 Appell hypergeometric function2F1 hypergeometric functionL tube length (m)n flow behavior indexP pressure drop (Pa)Q volumetric flow rate (m3.s−1)r tube radius (m)Rmax maximum radius of corrugated tube (m)Rmin minimum radius of corrugated tube (m)x axial coordinate (m)

Appendix A. Flow rate of power-law fluid in cylindrical tube

In this appendix, we derive an analytical expression for thevolumetric flow rate in a cylindrical duct, with inner radius r andlength x, assuming a power-law flow. We apply the well-knowngeneral result, sometimes called the Weissenberg–Rabinowitschequation or Rabinowitsch–Mooney equation (Bird et al., 1987; LiuandMasliyah, 1998) which relates the flow rate Q and the shear stressat the tube wall τw for laminar flow of time-independent fluids in acylindrical tube. This equation, which can be derived considering thevolumetric flow rate through a differential annulus between r′ and r′+dr′, is given by (Bird et al., 1987; Carreau et al., 1997; Skelland,1967; Sochi, 2007; Sochi and Blunt, 2008)

Qπr3

=1τ3w

∫τw0τ2γ̇ dτ: ð33Þ

For power-law fluids

τ = Cγ̇n ð34Þ

and hence the strain rate is given by

γ̇ =τ1=n

C1=n : ð35Þ

On substituting this relation into Eq. (33), the following is obtained

Qπr3

=1

τ3wC1=n ∫

τw0τ2 + 1=ndτ ð36Þ

i.e.

Q =πr3

τ3wC1=n 3 + 1= nð Þ τ3 + 1=n

h iτw0

=πr3τ3 + 1 = n

w

τ3wC1=n 3 + 1 = nð Þ =

πr3τ1 = nw

C1=n 3 + 1= nð Þ :

ð37Þ

Since

τw =Pr2x

ð38Þ

we obtain

Q =πr3

C1=n 3 + 1 = nð ÞPr2x

� �1=n=

πnr3P1=n

C1=n 3n + 1ð Þr2x

� �1=n ð39Þ

=πnr4P1=n

2xC1=n 3n + 1ð Þ2xr

� �2xr

� �−1=nð40Þ

that is

Q =πnr4P1=n

2xC1=n 3n + 1ð Þ2xr

� �1−1=nð41Þ

as given by Eq. (2).

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